3.332 \(\int \frac{\tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \, dx\)

Optimal. Leaf size=20 \[ 2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1) \]

[Out]

2*Sqrt[x]*ArcTan[Sqrt[x]] - Log[1 + x]

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Rubi [A]  time = 0.0154955, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ 2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[Sqrt[x]]/Sqrt[x],x]

[Out]

2*Sqrt[x]*ArcTan[Sqrt[x]] - Log[1 + x]

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Rubi in Sympy [A]  time = 1.76436, size = 17, normalized size = 0.85 \[ 2 \sqrt{x} \operatorname{atan}{\left (\sqrt{x} \right )} - \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(atan(x**(1/2))/x**(1/2),x)

[Out]

2*sqrt(x)*atan(sqrt(x)) - log(x + 1)

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Mathematica [A]  time = 0.00634974, size = 20, normalized size = 1. \[ 2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcTan[Sqrt[x]]/Sqrt[x],x]

[Out]

2*Sqrt[x]*ArcTan[Sqrt[x]] - Log[1 + x]

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Maple [A]  time = 0.007, size = 17, normalized size = 0.9 \[ -\ln \left ( 1+x \right ) +2\,\arctan \left ( \sqrt{x} \right ) \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x^(1/2))/x^(1/2),x)

[Out]

-ln(1+x)+2*arctan(x^(1/2))*x^(1/2)

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Maxima [A]  time = 1.33564, size = 22, normalized size = 1.1 \[ 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(x))/sqrt(x),x, algorithm="maxima")

[Out]

2*sqrt(x)*arctan(sqrt(x)) - log(x + 1)

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Fricas [A]  time = 0.225387, size = 22, normalized size = 1.1 \[ 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(x))/sqrt(x),x, algorithm="fricas")

[Out]

2*sqrt(x)*arctan(sqrt(x)) - log(x + 1)

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Sympy [A]  time = 0.707571, size = 17, normalized size = 0.85 \[ 2 \sqrt{x} \operatorname{atan}{\left (\sqrt{x} \right )} - \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(atan(x**(1/2))/x**(1/2),x)

[Out]

2*sqrt(x)*atan(sqrt(x)) - log(x + 1)

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GIAC/XCAS [A]  time = 0.213661, size = 22, normalized size = 1.1 \[ 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) -{\rm ln}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(x))/sqrt(x),x, algorithm="giac")

[Out]

2*sqrt(x)*arctan(sqrt(x)) - ln(x + 1)